Cover over, at first, open-air swimming pool (4)
I believe the answer is:
lido
'open-air swimming pool' is the definition.
(a lido is an outdoor swimming pool)
'cover over at first' is the wordplay.
'cover' becomes 'lid' (lid is a kind of cover).
'at first' indicates taking the first letters.
The first letter of 'over' is 'o'.
'lid'+'o'='LIDO'
(Other definitions for lido that I've seen before include "Aquatic leisure centre" , "Public open swimming pool" , "Open-air swimming pool" , "Italian bathing place" , "bathing-beach" .)